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Molecular constants of axially symmetric molecules from simultaneous analysis of ground-state combination differences
JOI-KNAL
OF MOLECULAR
SPECTROSCOPY
24, 111-115
(1967)
Molecular Constants of Axially Symmetric Simultaneous Analysis of Ground-State Differences *t
Department
of
Physics,
Michigan
Slate
C’nicersil!/,
Euvt
Molecules from Combination
/,nn.sing,
.llichigan
&X?S
A single analytical expression which represents all grrjlltltl-state combi~ratir~t~ differences for an axially symmetric molerulc is giver1 in t crms of ISo , D’I;, D/, etc. A simultaneous least squares analysis of all ohserved grc,r~l~ti-s+~tc~ combination differences for a particrdar molcclde, Iwing a11 cspression of this type, sholdd lead to values of BO , Di” , Dd, etc., which arc self-consistent aud are t,he best vallles which may be ohtailled from N given set of data. A4t~ additional term may be added to obtain 9 iu the event that the normal dcgeneracy of the upper K’ = I = fl levels is remtwcd, ~3.g.. t)y C’c,ri~,lis rest,nance, by P-type doubliug, etc.
band of an MYe designate an individual transition in :L vit)l,ation--rc,t:itioli axially symmetric molecule by a set of symbols ‘xAJg(.J ) where K and .I :tr(: the quantum numbers for the ground state, and AK and AJ are the changes it) these quantum numbers for the t’ransition indicated. The symbols I’, (2, and I{ are often used to indicate the specific AK and AJ valuc~s of -1, 0, ant1 +l, respectively, e.g., ‘P,(J), “CMJ), etc. 11 general wmbination difference of such transitions ie ““‘AJ,,,(JI
-
AJ1) -
=‘A.J~&~
-
A./“).
(1)
For such a pair of transitions tc I have a common upper state and thus represetlt a genuine ground-state combination difference for axially symmetric molcculcs~ it is necessary that Kl = Kp , AK1 = AKz , and J1 = ./s , actually the J value of the common upper state, with AJI and AJ, standing in relation to each othcbr as (+l, --I), C+l, O), or (0, -1). Using the usual symbols for AJ = 0, fl, * Supported t Partly
by a grant from the National
from a thesis by William
ments for the Ph.D.
degree at Michigan
.$ Now at the Department
Science
E. Blass
of Physics,
Fountlatiotl.
submit ted in part ial fulfillment
State
University,
8t. Mary’s 111
College,
of the rcql~iro-
1!)63. Wilwrla,
Miuncsoia.
we obtain
differences
of the t)ypes ‘KR,i.J
-
1) -
“KP,(.J
~KJM.J
-
1‘) -
Y&(J),
‘“Q,(J)
-
*“P&J
+ l),
+ l),
where AK nmy tnkc on vnlucs - 1, 0, + 1 (wit,h symbols P, Q, Iz). Clearly these combination differences may be extended directly to higher values of AJ and AK if such txu1sitions are observed (e.g., from Raman specka, electric field spectra, ckc. 1.
Q’riting the transition frequency lKA.JKi.J) in t’erms of the quantum numbers changes in clunntum numbers) t’s , lit , Al, , K, J, AK, and AJ as indeJ)dCllt variables, one has :I complicated expression for the frequencies of the individual lines originating from the ground vihrat’ional sstate (1, 2). The transition frequency esprcssion given by Harnett and Edwards (2), for example, includes 24 terms t,hrough the fourth-order t,ransformed Hamilt80nian of *Amat and Sicken (J) . A rclnt8ivcly simple espression for the ground-st’atc combination differences (G.S.C.D.), hsed on the combination difference, l’:q. ( 1)) is obtained from t,he general frequency espression in Table II of Reference 2 (and is also good to t,hc fourth order ) . The genernlized combination difference expression may be writ,trn in :I form directly suitable for least squares fits, viz., (and
(;.S.(:.n.(A./,
, AJZ, K, .J) = AKA.Jl,iJ
= I&,(A, -
A~) -
Lb”( At -
- I&“( AZ -
A#<’
+ /I:“( A2 -
A1)J? +
At)
+ H:l”(Az’ -
-
-
%Jz,LJ
+ Ho“CA,” -
~1”) +
A:)k”
AJI)
+
...
- AtJ.?! .. (2)
...
+ ..‘, where Ai = (.I - AJ;)iJ + 1 - A.J,), with i = 1 and 2, respectively, for the two transitions involved ill t)hc G.S.C.D. We have chosen t’he specific f(JrnlS ./ AJ, etc. so that here .J is the quantum number for the common upper level. l;:quation (2) is displayed in this form to make obvious how to extend it to include higher order terms. This single expression simultaneously represents all possible observable genuine ground-state combination differences which may be obtained from irlfrarcd :tbsor@n spwt’ra of axially symmetric molecules, viz., “RK(.J - 1) - “&(.J,, “Q,(.J1 - “1’d.J + I), and QRK(.J - 1) - ‘PK( J + 1) from parallel bands; and “C.j,,.(.J) - “P,CJ + l), RR,(.J - 1) - “(II( HR,i.J - 1)
- ‘PdJ + l), “QA J) - ‘I’&/ + l), “&(.I - 1) - l’&( .I), and “Rd./ - I) - ?,c J + 1) from perpendicular bands. P’IWa given molecule, all observable combination differences of any or all OF the above types, for all available values of K and *I, from all observable absorp tion bands (originating from t’he ground state’) may t,hen be simult~aneonsl~ analyzed by the method of least squurcs using the single expression, l&l. ( 2). ()f coluw, in order to separate R. from D ;” it is rlecessary to irJclude G.S.C.D.‘s from more than one sub-band, i.e., with different values of K. We note that from each perpendicular burJd ( or compoJJcntC there exist two sets of R - P’s, and one from each parallel band or compoJJent. Since all thrw sets are equivalent, beirJg est8imat8es of the separatioJJ hctmeerl the same levels. all should be gathered, compared, and used as a single wighted :tvcragc to put into Eq. (3). Similarly, because normally G.S.C.D. (R, (2, K, .J) = C+.S.(‘.I). t Q, P, K, J + l), there exist’ four sets of these from each pcrpeJJdicular band OI and two sets from each pnr:dlel band t lr componerJt. Here t 00, all componenb, should be gathered, compared, and :I single weighted :~vcrxgc put, into ICc(. ( 2 1. Such an analysis mill result in values of I?,, , IloJ, I>$“, etc. which arc self-COIL &tent and the best which may he obtairJed from :I giveJJ set, of data. I’wcoJJdit,ioning t,he data as outlined here will JJot. clxmgc values of paramc~terw ohtairJc>d or the total weight, but will change the statistics of the fit? i.c., standard tleviations and simult,aneous corJfidence intervals, through ( :I) use of the correct JJumber of degrees of freedom and (h) reduced input errors because such an average of several estimates of a quaJJtity (e.g., an averaged 12 - I’ combiJJatIioJJ difference for a given level separation) is a bet’ter estimate of the quant,ity. ( The &mate will have a smaller deviation since ~.d.,,,(.~,~= s.d., 2/~.) ]n the past it has been customary (4’) to use OIJC or both of two app:treJJt Iy dist,inct, expressions for ground-state comhiJr:~tioJJ difirrcwccs of axially syncmetric molecules, representable symbolically as lx&,( J I -
.‘KP,(
J + 1) = Ah&( .J) - ‘“J’d = 2(B,, -
K%‘;“)i.J
.J + 1 I + 1) -
IJh,“(J
+ I ):;
( :<)
:111d
AK’KK(J
-
1) -
““P&J = (4!Bo -
+
I) K”n;:“,
-
Wo“]( .J + f’? I -
XJ&“c .J + 1” ).:’
(4)
The eq~p&on we have set down in &I. (2) corJt:JiJJs both E(ls. ( 3) :ltJ(! (1 I. [Equations ( 3) and (4) may he obtained from &. i 2 ) f’m- :Jpproprint~e particular choices of ilJI and AJr .] The apparent distiJJctness of Ells. ( 3) and (1 1 has somet,imes led to solving each set separately aJJd averaging t,he resulting values 01 & , DoJ, alld 0;” (or, if only a few of one type and more of t,he other type wre
BLASS AND EDWARDS
114
found, one might be led to neglect t’he lesser number which were not obviously compatible with t’he many). In addition, it has been rather common practice to divide Eq. (3) by (J + 1) and Eq. (4) by (J + .S$j and plot the resulting fixed K series for each observable J value vs. (.J + 1 j’ or iJ + ?.4)2 in order to get a helpful over-all picture of the dat,a. Unfortunately, it has also been common practice to fit, these series of ploiked points either graphically or by the method of least, squares. This practice may lead to two undesirable results: first,, a (l~resunlably uilintentional) variable de~veightiljg of the combination differences by the factor J + 1 or J + ;2$, respectively, and in addition, fitting these series separately may lead to different, and therefore inconsistent, values for BQ and L)oJ from each series. We recommend fitting the combinatian differences simultaneously by Ieast squares to a single equation such as ours, incorporating any intentional weighting in this data if the character and number of the observed combination differences warrant’s it. This method will be applied in the following article to CHD3 . CORIOLIS
RESONANCE,
P-TYPE DOUBLING,
ETC.
True ground-state combination differences are not affected by perturbat,ions involving t,he upper levels of vil~ration-rotation transitions. As a result, the etc., which one obtains from gro~illcl-st,ate colnbi~~ation values of B. , DtIJ, DOE", differences cm be expected to be insensitive to ~~erturbat,ions of the upper level provided these perturbat’iotls are not so serious as to render doubtful the assignment of the lines used. An exception may occur when the transitions involved in a combination difference terminate on a normally degenerate set of upper levels which can be split by a perturbation. It is possible, in this case, that the perturbation may lift the degeneracy, and thus the various transitions terminating on these levels (which have cquai energies in the absence of a perturbation) would now terminatc! on ho levels of different, energy. Esampies of such an occurrence arc provided by Coriolis resonance, &type do~lblit~g, and giant. &typc doubling which lift the degeneracy of the K’ = 4 = fl levels (i.e., Kd = +t levels) in the upper stat.e. In every case, t,he I2 - P c(~rnbi~l~~t.ion differences, even for K” = 0, are stiI1 genuine ground-state combination differences. However, the R - Q and Q - P combinatjion difierenccls are now no longer “true” ground-state combination differences because the “Qo(J) transitions terminate on one upper level of t,he “formerly” degenerate pair while the RRo( J ) and RP,( J) transitions terminate on the other. However, using Eq. (2), it is still possible to analyze simultaneously all ground-state combination differences except the particular RO - Q. and Q. - PO differences which are affected. Moreover, if one is confident of t’he form of the perturbation, me may sometimes include a term to handle even these; e.g., t2his may be done by adding to IQ. (3) a term such as jiogq J(J + 1)
SIMULTANEOUW
COMBINATION
T)IFFERESCES
1 1.5
h . ( &J1’ - bJ2’), where & = the Kronecker 6 taking t,he value 1 when K = 0 nrld 0 for all other values of K, for the hand in question. We have followed such a procedure in analyzing vii + v4 of C’HnF C1, ;i i. I~ECEIVED: ;\larch
22, 1967 REFERENCW
1. W. E. BL.ISS, thesis, Michigan State University-: 1963. 2. T. L. BURNETT AND T. H. EDIV-~RDS,J. Jlol. Specfry. 20, 347 (1966). 3. RI. GOLDSMITH, Ct. AMAT, .\ND II. H. NIELSEN, J. (,‘hem. Phys. 24, 1178 (1956); G. AM-IT, M. GOLDSMITH, AND H. II. NIELSEN, J. (Ihem. Phys. 27, 838 (1957); G. Anur .IND H. H.H NIELSEN, J. Chew Phys. 27, 845 1195i); G. AMAT AXD H. H. NIELSEX, J. Chenl. Phys. 29, 665 (195X); G. AM.IT :\ND H. H. NIELSEN, J. Chem. Phys. 36, 1859 (1962); M. L. GHENIER-BESSON, G. AM;IT, &*NDII. II. NIELSEN, .I. C’h.em. Phys. 36, 3454 (lW2). 4. W. S. BENEDICT, E. K. PLYLER, .YNDE. D. TIIJIVELL, .I. Kcs. :\-all. 13ur. Std. (ri.S.) 61, 123 (1958). 6. W. E. Btass AND T. H. EDWARDS (to be published).
OF MOLECULAR
SPECTROSCOPY
24, 111-115
(1967)
Molecular Constants of Axially Symmetric Simultaneous Analysis of Ground-State Differences *t
Department
of
Physics,
Michigan
Slate
C’nicersil!/,
Euvt
Molecules from Combination
/,nn.sing,
.llichigan
&X?S
A single analytical expression which represents all grrjlltltl-state combi~ratir~t~ differences for an axially symmetric molerulc is giver1 in t crms of ISo , D’I;, D/, etc. A simultaneous least squares analysis of all ohserved grc,r~l~ti-s+~tc~ combination differences for a particrdar molcclde, Iwing a11 cspression of this type, sholdd lead to values of BO , Di” , Dd, etc., which arc self-consistent aud are t,he best vallles which may be ohtailled from N given set of data. A4t~ additional term may be added to obtain 9 iu the event that the normal dcgeneracy of the upper K’ = I = fl levels is remtwcd, ~3.g.. t)y C’c,ri~,lis rest,nance, by P-type doubliug, etc.
band of an MYe designate an individual transition in :L vit)l,ation--rc,t:itioli axially symmetric molecule by a set of symbols ‘xAJg(.J ) where K and .I :tr(: the quantum numbers for the ground state, and AK and AJ are the changes it) these quantum numbers for the t’ransition indicated. The symbols I’, (2, and I{ are often used to indicate the specific AK and AJ valuc~s of -1, 0, ant1 +l, respectively, e.g., ‘P,(J), “CMJ), etc. 11 general wmbination difference of such transitions ie ““‘AJ,,,(JI
-
AJ1) -
=‘A.J~&~
-
A./“).
(1)
For such a pair of transitions tc I have a common upper state and thus represetlt a genuine ground-state combination difference for axially symmetric molcculcs~ it is necessary that Kl = Kp , AK1 = AKz , and J1 = ./s , actually the J value of the common upper state, with AJI and AJ, standing in relation to each othcbr as (+l, --I), C+l, O), or (0, -1). Using the usual symbols for AJ = 0, fl, * Supported t Partly
by a grant from the National
from a thesis by William
ments for the Ph.D.
degree at Michigan
.$ Now at the Department
Science
E. Blass
of Physics,
Fountlatiotl.
submit ted in part ial fulfillment
State
University,
8t. Mary’s 111
College,
of the rcql~iro-
1!)63. Wilwrla,
Miuncsoia.
we obtain
differences
of the t)ypes ‘KR,i.J
-
1) -
“KP,(.J
~KJM.J
-
1‘) -
Y&(J),
‘“Q,(J)
-
*“P&J
+ l),
+ l),
where AK nmy tnkc on vnlucs - 1, 0, + 1 (wit,h symbols P, Q, Iz). Clearly these combination differences may be extended directly to higher values of AJ and AK if such txu1sitions are observed (e.g., from Raman specka, electric field spectra, ckc. 1.
Q’riting the transition frequency lKA.JKi.J) in t’erms of the quantum numbers changes in clunntum numbers) t’s , lit , Al, , K, J, AK, and AJ as indeJ)dCllt variables, one has :I complicated expression for the frequencies of the individual lines originating from the ground vihrat’ional sstate (1, 2). The transition frequency esprcssion given by Harnett and Edwards (2), for example, includes 24 terms t,hrough the fourth-order t,ransformed Hamilt80nian of *Amat and Sicken (J) . A rclnt8ivcly simple espression for the ground-st’atc combination differences (G.S.C.D.), hsed on the combination difference, l’:q. ( 1)) is obtained from t,he general frequency espression in Table II of Reference 2 (and is also good to t,hc fourth order ) . The genernlized combination difference expression may be writ,trn in :I form directly suitable for least squares fits, viz., (and
(;.S.(:.n.(A./,
, AJZ, K, .J) = AKA.Jl,iJ
= I&,(A, -
A~) -
Lb”( At -
- I&“( AZ -
A#<’
+ /I:“( A2 -
A1)J? +
At)
+ H:l”(Az’ -
-
-
%Jz,LJ
+ Ho“CA,” -
~1”) +
A:)k”
AJI)
+
...
- AtJ.?! .. (2)
...
+ ..‘, where Ai = (.I - AJ;)iJ + 1 - A.J,), with i = 1 and 2, respectively, for the two transitions involved ill t)hc G.S.C.D. We have chosen t’he specific f(JrnlS ./ AJ, etc. so that here .J is the quantum number for the common upper level. l;:quation (2) is displayed in this form to make obvious how to extend it to include higher order terms. This single expression simultaneously represents all possible observable genuine ground-state combination differences which may be obtained from irlfrarcd :tbsor@n spwt’ra of axially symmetric molecules, viz., “RK(.J - 1) - “&(.J,, “Q,(.J1 - “1’d.J + I), and QRK(.J - 1) - ‘PK( J + 1) from parallel bands; and “C.j,,.(.J) - “P,CJ + l), RR,(.J - 1) - “(II( HR,i.J - 1)
- ‘PdJ + l), “QA J) - ‘I’&/ + l), “&(.I - 1) - l’&( .I), and “Rd./ - I) - ?,c J + 1) from perpendicular bands. P’IWa given molecule, all observable combination differences of any or all OF the above types, for all available values of K and *I, from all observable absorp tion bands (originating from t’he ground state’) may t,hen be simult~aneonsl~ analyzed by the method of least squurcs using the single expression, l&l. ( 2). ()f coluw, in order to separate R. from D ;” it is rlecessary to irJclude G.S.C.D.‘s from more than one sub-band, i.e., with different values of K. We note that from each perpendicular burJd ( or compoJJcntC there exist two sets of R - P’s, and one from each parallel band or compoJJent. Since all thrw sets are equivalent, beirJg est8imat8es of the separatioJJ hctmeerl the same levels. all should be gathered, compared, and used as a single wighted :tvcragc to put into Eq. (3). Similarly, because normally G.S.C.D. (R, (2, K, .J) = C+.S.(‘.I). t Q, P, K, J + l), there exist’ four sets of these from each pcrpeJJdicular band OI and two sets from each pnr:dlel band t lr componerJt. Here t 00, all componenb, should be gathered, compared, and :I single weighted :~vcrxgc put, into ICc(. ( 2 1. Such an analysis mill result in values of I?,, , IloJ, I>$“, etc. which arc self-COIL &tent and the best which may he obtairJed from :I giveJJ set, of data. I’wcoJJdit,ioning t,he data as outlined here will JJot. clxmgc values of paramc~terw ohtairJc>d or the total weight, but will change the statistics of the fit? i.c., standard tleviations and simult,aneous corJfidence intervals, through ( :I) use of the correct JJumber of degrees of freedom and (h) reduced input errors because such an average of several estimates of a quaJJtity (e.g., an averaged 12 - I’ combiJJatIioJJ difference for a given level separation) is a bet’ter estimate of the quant,ity. ( The &mate will have a smaller deviation since ~.d.,,,(.~,~= s.d., 2/~.) ]n the past it has been customary (4’) to use OIJC or both of two app:treJJt Iy dist,inct, expressions for ground-state comhiJr:~tioJJ difirrcwccs of axially syncmetric molecules, representable symbolically as lx&,( J I -
.‘KP,(
J + 1) = Ah&( .J) - ‘“J’d = 2(B,, -
K%‘;“)i.J
.J + 1 I + 1) -
IJh,“(J
+ I ):;
( :<)
:111d
AK’KK(J
-
1) -
““P&J = (4!Bo -
+
I) K”n;:“,
-
Wo“]( .J + f’? I -
XJ&“c .J + 1” ).:’
(4)
The eq~p&on we have set down in &I. (2) corJt:JiJJs both E(ls. ( 3) :ltJ(! (1 I. [Equations ( 3) and (4) may he obtained from &. i 2 ) f’m- :Jpproprint~e particular choices of ilJI and AJr .] The apparent distiJJctness of Ells. ( 3) and (1 1 has somet,imes led to solving each set separately aJJd averaging t,he resulting values 01 & , DoJ, alld 0;” (or, if only a few of one type and more of t,he other type wre
BLASS AND EDWARDS
114
found, one might be led to neglect t’he lesser number which were not obviously compatible with t’he many). In addition, it has been rather common practice to divide Eq. (3) by (J + 1) and Eq. (4) by (J + .S$j and plot the resulting fixed K series for each observable J value vs. (.J + 1 j’ or iJ + ?.4)2 in order to get a helpful over-all picture of the dat,a. Unfortunately, it has also been common practice to fit, these series of ploiked points either graphically or by the method of least, squares. This practice may lead to two undesirable results: first,, a (l~resunlably uilintentional) variable de~veightiljg of the combination differences by the factor J + 1 or J + ;2$, respectively, and in addition, fitting these series separately may lead to different, and therefore inconsistent, values for BQ and L)oJ from each series. We recommend fitting the combinatian differences simultaneously by Ieast squares to a single equation such as ours, incorporating any intentional weighting in this data if the character and number of the observed combination differences warrant’s it. This method will be applied in the following article to CHD3 . CORIOLIS
RESONANCE,
P-TYPE DOUBLING,
ETC.
True ground-state combination differences are not affected by perturbat,ions involving t,he upper levels of vil~ration-rotation transitions. As a result, the etc., which one obtains from gro~illcl-st,ate colnbi~~ation values of B. , DtIJ, DOE", differences cm be expected to be insensitive to ~~erturbat,ions of the upper level provided these perturbat’iotls are not so serious as to render doubtful the assignment of the lines used. An exception may occur when the transitions involved in a combination difference terminate on a normally degenerate set of upper levels which can be split by a perturbation. It is possible, in this case, that the perturbation may lift the degeneracy, and thus the various transitions terminating on these levels (which have cquai energies in the absence of a perturbation) would now terminatc! on ho levels of different, energy. Esampies of such an occurrence arc provided by Coriolis resonance, &type do~lblit~g, and giant. &typc doubling which lift the degeneracy of the K’ = 4 = fl levels (i.e., Kd = +t levels) in the upper stat.e. In every case, t,he I2 - P c(~rnbi~l~~t.ion differences, even for K” = 0, are stiI1 genuine ground-state combination differences. However, the R - Q and Q - P combinatjion difierenccls are now no longer “true” ground-state combination differences because the “Qo(J) transitions terminate on one upper level of t,he “formerly” degenerate pair while the RRo( J ) and RP,( J) transitions terminate on the other. However, using Eq. (2), it is still possible to analyze simultaneously all ground-state combination differences except the particular RO - Q. and Q. - PO differences which are affected. Moreover, if one is confident of t’he form of the perturbation, me may sometimes include a term to handle even these; e.g., t2his may be done by adding to IQ. (3) a term such as jiogq J(J + 1)
SIMULTANEOUW
COMBINATION
T)IFFERESCES
1 1.5
h . ( &J1’ - bJ2’), where & = the Kronecker 6 taking t,he value 1 when K = 0 nrld 0 for all other values of K, for the hand in question. We have followed such a procedure in analyzing vii + v4 of C’HnF C1, ;i i. I~ECEIVED: ;\larch
22, 1967 REFERENCW
1. W. E. BL.ISS, thesis, Michigan State University-: 1963. 2. T. L. BURNETT AND T. H. EDIV-~RDS,J. Jlol. Specfry. 20, 347 (1966). 3. RI. GOLDSMITH, Ct. AMAT, .\ND II. H. NIELSEN, J. (,‘hem. Phys. 24, 1178 (1956); G. AM-IT, M. GOLDSMITH, AND H. II. NIELSEN, J. (Ihem. Phys. 27, 838 (1957); G. Anur .IND H. H.H NIELSEN, J. Chew Phys. 27, 845 1195i); G. AMAT AXD H. H. NIELSEX, J. Chenl. Phys. 29, 665 (195X); G. AM.IT :\ND H. H. NIELSEN, J. Chem. Phys. 36, 1859 (1962); M. L. GHENIER-BESSON, G. AM;IT, &*NDII. II. NIELSEN, .I. C’h.em. Phys. 36, 3454 (lW2). 4. W. S. BENEDICT, E. K. PLYLER, .YNDE. D. TIIJIVELL, .I. Kcs. :\-all. 13ur. Std. (ri.S.) 61, 123 (1958). 6. W. E. Btass AND T. H. EDWARDS (to be published).